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Tag Archives: modules
Commutative Algebra 10
Algebras Over a Ring Let A be any ring; we would like to look at Amodules with a compatible ring structure. Definition. An –algebra is an module , together with a multiplication operator such that becomes a commutative ring (with 1); multiplication … Continue reading
Posted in Advanced Algebra
Tagged algebras, generated submodules, homomorphism, modules, quotient modules, rings, submodules
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Commutative Algebra 9
Direct Sums and Direct Products Recall that for a ring A, a sequence of Amodules gives the Amodule where the operations are defined componentwise. In this article, we will generalize the construction to an infinite collection of modules. Throughout this article, let denote … Continue reading
Posted in Advanced Algebra
Tagged direct products, direct sums, modules, rings, universal properties
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Commutative Algebra 8
Generated Submodule Since the intersection of an arbitrary family of submodules of M is a submodule, we have the concept of a submodule generated by a subset. Definition. Given any subset , let denote the set of all submodules of M containing … Continue reading
Posted in Advanced Algebra
Tagged free modules, generated submodules, homomorphism, modules, quotient modules, rings, submodules
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Commutative Algebra 7
Modules Having dipped our toes into algebraic geometry, we are back in commutative algebra. Next we would like to introduce “linear algebra” over a ring A. Most of the proofs should pose no difficulty to the reader so we will … Continue reading
Posted in Advanced Algebra
Tagged ideals, linear algebra, module homomorphism, modules, rings, submodules
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Idempotents and Decomposition
Let R be a general ring, not necessarily commutative. An element x∈R is said to be idempotent if x2 = x. Note An endomorphism f of an Rmodule M (i.e. ) is an idempotent if and only if f is a projection, i.e. M = ker(f) ⊕ im(f) and f … Continue reading
Posted in Notes
Tagged blocks, idempotents, indecomposable modules, modules, primitive idempotents
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Tensor Product over Noncommutative Rings
Following the earlier article on tensor products of vector spaces, we will now look at tensor products of modules over a ring R, not necessarily commutative. It turns out we have to distinguish between left and right modules now. Indeed recall … Continue reading
Posted in Notes
Tagged bimodules, hom functor, leftexact, modules, rightexact, tensor products
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Hom Functor
Fret not if you’re unfamiliar with the term functor; it’s a concept in category theory we will use implicitly without delving into the specific definition. This topic is, unfortunately, a little on the dry side but it’s a necessary evil to get … Continue reading
Posted in Notes
Tagged bimodules, hom functor, left modules, leftexact, modules, right modules
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